Why Legs have Three Joints

John Nagle

Introduction

Look at the hind leg of a running animal, such as a horse or dog. The leg is
divided into three parts of roughly equal length. Why three joints? Two joints
would be enough to place the foot in any desired position. So why the extra
joint? There must be some big advantage, or legs wouldn't have evolved that
way.

There are three joints because a leg with three joints can climb a slippery
hill that a leg with two joints can't. This paper explains why this is so. This
fact has implications for legged robots and animated legged creatures.

The reader is assumed to be familiar with Marc Raibert's work, in particular
his book, "Legged Robots that Balance"[1]
, as well as his paper with Jessica Hodgins on kangaroo locomotion.[2] Raibert controls articulated legs so
they behave like the straight leg of the planar hopper. I have some
improvements on that approach. This is an early note on some new work.

The anti-slip controller described in the author's previous paper[3] limits the torque applied to the balance
actuator. It does this to limit the transverse forces between the foot and the
ground, because that's what causes slipping. On slippery surfaces, this limit
prevents large balance torques from being applied during the stance phase,
which in turn limits the motions possible on slippery surfaces. This is OK, but
limiting, especially during hill climbing, where the torque limits are rather
low due to the hill angle.

With the straight-legged hopper, the thrust forces exerted by the hopping
actuator are always aligned with the leg. With an articulated leg, there are
more options. Raibert throws these options away, by always placing the knee so
as to direct the thrust forces through a fixed point, usually the system center
of gravity. That's not necessary, and there are real advantages to using those
extra joints.

Because he assumes very high ground friction, Raibert doesn't have to worry
about limiting balance torques. Once limited ground friction is accepted,
you're forced to more realistic control approaches.

Example - a kangaroo leg

This is a kangaroo-type leg structure. The tail has been omitted, which is why
it looks front-heavy. In this drawing, an articulated leg is linked to an
equivalent mechanism like that of the planar hopper. You probably wouldn't
actually build something like that, with both the articulated leg and the
straight leg linkages tied together. It's a convenient way to help think about
the motions involved.

There's an additional degree of freedom here. A second sliding joint is added
to the hopper-like linkage, allowing forward/back motion of the point where the
thrust is applied to the body. This provides an additional degree of freedom
which maps directly to the moment applied to the body mass by the leg.

The next set of drawings shows how this works.

Here we see the leg in three different positions. The body and foot are in the
same position for all three drawings, but the limbs have been moved to change
the line of action of the thrust force. In the left drawing, the thrust force
is being applied so as to produce a pitch-up rotation of the body. The middle
drawing shows a neutral situation, where the thrust force is being applied
through the center of gravity, thus imparting no rotational forces to the body.
The right drawing shows the thrust force being applied to produce a pitch-down
situation. Notice that with this structure, we can change the moment applied to
the body without changing the foot position.

The result of this action is to apply rotational forces to the body, so it
does roughly the same thing as the balance actuator does in the planar hopper.
But it does it with smaller transverse forces at the foot/ground contact point.
Thus, bigger balancing torques can be applied in this way than via the balance
actuator alone. This allows more aggressive hill-climbing and operation on
slippery surfaces.

The next step is to take away the hopper linkage, leaving only the articulated
leg.

The three drawings illustrate the line of thrust being applied in front of,
through, and behind, the center of gravity. But now this function is performed
by servoing the rotational joints, rather than via a linkage. Detailed control
strategies will be covered in a later paper.

Conclusion

So that's what the extra joint in legs does. Observation of photographs of
real animals on hills indicates that this is how real animals use their legs.
It's especially clear when you look at photos of animals with big hindquarters,
like horses, going up and down hills. So this technique will produce more
realistic animations. Check out Disney's "The Lion King". The
computer-generated sequence of the herd stampede looks fine until the herd
starts descending a steep hill. Downhill, the running looks totally
unrealistic. It looks like fish going over a waterfall. The animators seem to
have just used a gait cycle, even though that's totally inappropriate to the
movement. The technology described here is going to be able to do a much better
job at that sort of thing. Perhaps up to the quality of the live action in
"The Man from Snowy River".